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s51 - Solutions to Homework#5 Chapter 22 54 Picture the...

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Solutions to Homework #5 Chapter 22 54. Picture the Problem: Two current-carrying wires each make a contribution to the magnetic field at point P. Strategy: The magnetic field at point P is the vector sum of the magnetic fields 1 2 and B B r r from the two wires. Use equation 22-9 to find the magnitude and the Right-Hand Rule to find the direction of each field. Then add the two fields as vectors in order to find the magnitude and direction of the net magnetic field at P. From the diagram we can see that 1 0.050 m, r = ( 29 ( 29 ( 29 2 2 2 1 0.050 m 0.050 m 2 0.050 m 2 , r r = + = = and that 2 B r points 45° below the − x axis. Solution: 1. Add the two vectors using equation 22-9 and the RHR: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 0 1 0 2 0 2 1 1 2 1 7 6 6 ˆ ˆ ˆ ˆ ˆ ˆ cos45 sin 45 2 2 2 2 2 2 4 10 T m A 4.0 A ˆ ˆ ˆ ˆ ˆ 3.0 A 4.0 10 T 8.0 10 T 2 0.050 m 2 I I I I r r r μ μ μ π π π π π - - - = + = + - - = + - - = - + = - B B B x y x x y x B x x y x y r r r r 2. Find the magnitude of : B r ( 29 ( 29 2 2 6 6 6 4.0 10 T 8.0 10 T 8.9 10 T 8.9 T B μ - - - = + - = = 3. Find the direction of : B r 6 1 1 6 8.0 10 T tan tan 63 below the dashed line to the right of 4.0 10 T y x B P B θ - - - - - = = = Insight: 2 B is smaller than 1 B by a factor of 2 due to the farther distance of 2 I from P, but is larger by a factor of 4 3 due to the larger current. The two effects nearly cancel out, making 2 B only 5.7% smaller than 1 . B 56. Picture the Problem: The number of turns in a solenoid is doubled, and at the same time its length is doubled. Strategy: Use 0 N B I L μ � � = � � � � (equation 22-12) to predict the change in the magnetic field that would result from the indicated changes in the solenoid. Solution: 1. (a) Equation 22-12 shows that doubling both the number of turns N and the length L of a solenoid will have no effect upon the magnetic field produced by the solenoid. We conclude that the magnetic field within the solenoid would stay the same. 2. (b) The best explanation is III . The magnetic field remains the same because the number of turns per length is unchanged. Statement I ignores the change in L , and statement II ignores the change in N . Insight: If the current in the solenoid were also doubled, the magnetic field strength would double. 82. Picture the Problem: A square loop of current and a straight current-carrying wire exert forces on each other.
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